Height Ridge Computation and Filtering for Visualization

Abstract

Motivated by the growing interest in the use of ridges in scientific visualization, we analyze the two height ridge definitions by Eberly and Lindeberg. We propose a raw feature definition leading to a superset of the ridge points as obtained by these two definitions. The set of raw feature points has the correct dimensionality, and it can be narrowed down to either Eberly's or Lindeberg's ridges by using Boolean filters which we formulate. While the straight-forward computation of height ridges requires explicit eigenvalue calculation, this can be avoided by using an equivalent definition of the raw feature set, for which we give a derivation. We describe efficient algorithms for two special cases, height ridges of dimension one and of co-dimension one. As an alternative to the aforementioned filters, we propose a new criterion for filtering raw features based on the distance between contours which generally makes better decisions, as we demonstrate on a few synthetic fields, a topographical dataset, and a fluid flow simulation dataset. The same set of test data shows that it is unavoidable to use further filters to eliminate false positives. For this purpose, we use the angle between feature tangent and slope line as a quality measure and, based on this, formalize a previously published filter.

Ridge–Valley graphs: Combinatorial ridge detection using Jacobi sets

“…For a detailed discussion on some of the more popular approaches, we refer the reader to [16,2,17,18,19]. It has been previously noted that all of the ridges and valleys in the height ridge definition, must be points where the gradient of the height function and the gradient of the magnitude are aligned [17,20,19,13], but the subtle consequences of this observation have not been examined, which is one of the results of this paper. In their recent paper, Sadlo and Peikert [19] take advantage of this fact to extract "raw features," which are then filtered to produce ridges and valleys.…”

“…Numerical level set extraction will produce 1-dimensional loops, however we show in the course of this paper, that these features have a more complex structure and will in fact cross at every critical point. Thus without some sort of heuristic or connecting procedure the ridges in [19] will be disconnected at every critical point, which they admit is a difficulty. The algorithm presented in this paper avoids this shortcoming by employing a combinatorial algorithm, which is guaranteed to connect ridges and valleys through critical points.…”

“…Thus, when the behavior of the function f is dominated by second and lower order effects, one would expect the two definitions to produce similar results. In practice, Sadlo and Peikert [19] found that the Jacobi ridges were slightly, qualitatively preferable.…”

Features in Continuous Parallel Coordinates

“…Thus, known scalar field features might occur, cf. [7,9,21]. Additionally, the investigation of the feature's relations is important to facilitate the visual analysis of data by CPCs.…”

“…Therefore, the density map from a saddle critical point onto the CPC does not reveal a discontinuity, but a local maximum curve, a so-called ridge curve (cf. [21]). This is an unexpected result because it means that discontinuity features completely disappear within a corresponding CPC.…”